What does it mean when Chopin says "he was reminded that she was nameless"?
The story begins with a flashback description of how Desiree was found as a toddler at the side of the road, right at the gate to the Valmonde's property. Madame Valmonde, who was childless, adopted Desiree and considered her a gift from Providence. When Desiree grew to womanhood, she attracted the attention and head-over-heels love of Armand Aubigny. Desiree's father became practical when Aubigny proposed. Monsieur Valmonde reminded Armand that Desiree "was nameless." That is, he clarified for Armand that Desiree's birth family and genetic heritage were unknown. For someone like Armand, who it becomes clear later, has definite opinions about race, this should have served as a warning that he should take care about marrying Desiree. But Armand was so infatuated that he disregarded that warning and married her anyway. He thought at the time, the story implies, that his name—one respected for generations—would be sufficient for both of them.
However, when Desiree's baby is born and has physical characteristics suggesting he has black heritage, Armand is able to "blame" the child's mixed race on Desiree. Although Chopin doesn't say so, she leaves enough ambiguity in the story to suggest that Armand could have known that he himself had black ancestry. After all, he was eight years old when his mother died—old enough to remember her looks and to wonder why she had never been brought to their plantation. Armand's mother's hope was that Armand would never know she was black, but when he reads her letter at the end of the story, we are not told if that was the first time he had seen it.
One way to read the story, although perhaps not the most common way, is that Armand chose a wife who "was nameless" so that, if his offspring showed black characteristics, he could attribute that to the mother rather than reveal his own heritage. The most common reading is that Armand truly does believe Desiree is the "carrier" of the black heritage until he finds the letter from his mother at the end of the story. Either way, the fact that Desiree's heritage is unknown allows the story to unfold as it does.
As "Desire's Baby" begins, the reader is introduced to Desire. As a baby, she is found on the side of the road by the Valmounde family. A wealthy family with no children of their own they take her in and adopt her. Years later when she has grown into a beauty, she catches the eye of a neighboring plantation owner, Armand Aubigny. As a part of a wealthy family, Armand has a wealthy name- one that has been passed down from generation to generation. When they first get married, Armand's father warns him that Desire has no such name. Since she was adopted, no one knows who her family is, or where they come from. At the time he is in love and doesn't worry about it. He dismisses his father's warning saying that he will give Desire his name.
A sweet thought that comes back to haunt him once their baby is born. When they realize that the baby has color to him they realize there must be some Black ancestry in their family. Since she doesn't "have a name" Armand believes it must be her fought and her family. The true irony exists when we realize that it is actually Black ancestry.